Quantum Mechanics and the Continuum Limit of an Emergent Geometry

Recent advances in emergent geometry have identified a new class of models that represent spacetime as the graph obtained as the ground state of interacting Ising spins. These models have many desirable features, including have stable excitations possessing many of the characteristics of quantum particles. We analyze the dynamics of such excitations, including a detailed treatment of the edge states not previously addressed. Using a minimal prescription for the interaction of defects we numerically investigate approximate bounds to the speed of propagation of such a `particle'. We discover, using numerical simulations, that there may be a Lieb-Robinson bound to propagation that could point the way to how a causal structure could be accommodated in this class of emergent geometry models.